Questions tagged [measurable-functions]
For questions about measurable functions.
1,290 questions
1 vote
0 answers
58 views
Representation of this indicator function.
Let $(S,\Sigma)$ be a measurable space, $f_1,\cdots,f_n:S\to\Bbb R$ be measurable functions, and $f:=(f_1,\cdots,f_n)$. For $A\in \mathcal F:=\{f^{-1}(B)\mid B\in\mathcal B(\Bbb R^n)\}$, I want to ...
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1 vote
0 answers
50 views
$\sigma(f(X))\subseteq \sigma(X)$ and $\sigma(f(X))=\sigma(X)$ if and only if $f$ is an injective or bijective map?
Let $(\Omega, \mathcal{A})$ be a measurable space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ a measurable function and $f:(\mathcal{X}, \mathcal{F})\rightarrow (\mathcal{Z}, \...
- 742
0 votes
0 answers
78 views
Is this function measurable on $\Bbb{R}^2$?
Let $f$ be a function on $\Bbb R^2 \setminus S^1$ defined as, $f(x)=0$ if $|x|<1$ and $f(x)=1$ if $|x|>1$. Now we define, $F$ on $\Bbb R^2$ as $F(x)=f(x)$ if $x \in \Bbb R^2 \setminus S^1$ and ...
- 2,401
0 votes
0 answers
78 views
$f(x)$, $g(x)$ and $h(x)$ are assumed to be integrable, is there a simpler set of assumptions I can use instead?
Background I'm working on formalising a theorem in lean. ...
- 1,050
0 votes
1 answer
33 views
Existence of a convergence sequence in weak-star topology
In a book concerning calculus of variations written by Giusti, I read the following " Let $Q$ be a cube. For every $\lambda\in [0,1]$ there exists a sequence $\chi_h$ of characteristic functions ...
- 928
8 votes
1 answer
380 views
If two measurable functions have a.e. equal preimages, are they a.e. equal?
Let $(X,\mathcal{A},\mu)$ be a measure space, and let $f,g: X\to X$ be two measurable functions. Assume that the preimages of measurable sets are a.e. the same for $f$ and $g$, that is, $$ \mu(f^{-1}(...
0 votes
1 answer
49 views
Correspondence between Bernstein sets
I am considering a Bernstein set $A$ and its complementary $A^c$ in the reals. I understand that since both are of the same cardinality it is possible to make a bijection $f: A \rightarrow A^c$. The ...
- 3
3 votes
1 answer
215 views
Dependence of the statement that $f=g$ holds $\mu$-almost everywhere on the $\sigma$-algebra
Motivated by my attempts to understand conditional expectations I have come to ask myself a question regarding statements in measure theory that hold $\mu$-almost everywhere. So in conditional ...
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